Are all connected manifolds homogeneous

Solution 1:

Yes, any connected topological manifold $X$ of arbitrary dimension $n$ is homogeneous .

1) The crucial lemma is that given two points $a,b\in \mathbb B^{\circ}$ in the interior of a closed ball $\mathbb B \subset \mathbb R^n$, there exists a homeomorphism $f: \mathbb B\to \mathbb B$ which is the identity on $\partial \mathbb B$ and such that $f(a)=b$.

2) It then follows that if you fix any point $x_0 \in X$, then the set of points $y\in X$ that can be written $y=F(x_0)$ for some homeomorphism $F:X\to X$ is both open and closed, hence is equal to $X$.
Hence $X$ is homogeneous.
(By the way, an obvious modification of the proof shows that the analogous result is also true for a differential manifold: its diffeomorphisms act transitively on the manifold)

Edit: a fishy image
Let me give a physical model which might help visualize the lemma in 1) ( a totally rigorous and amazingly crisp proof is given in t.b.'s great comment).
Imagine you have a spherical fishbowl completely filled with water and a goldfish sitting somewhere in it.
The lemma says that you can send the goldfish to any preassigned place in the bowl by skilfully (!) shaking the bowl.